The Kardar-Parisi-Zhang (KPZ) equation [1]} is a paradigm for nonequilibrium systems in which generic scale invariance (GSI) occurs [2]}, [3]}. It is the representative of an universality class for space-time criticality — characterized by how the amplitude of the field fluctuations scales in space and time and how these fluctuations are statistically distributed [4]}, [5]}, [6]}, [7]}, [8]}, [9]} — in absence of parameter tuning, which describes a wide variety of systems with strong correlations. Some of these systems include turbulent liquid crystals [10]}, stochastic hydrodynamics [11]}, colloidal aggregation [12]}, reaction-diffusion systems [13]}, random geometry [14]}, [15]}, active matter [16]}, thin films [17]}, superfluidity [18]}, quantum entanglement [19]}, electronic fluids [20]}, and both integrable and non-integrable quantum spin chains [21]}, [22]}, [23]}. Specifically, the KPZ equation describes the evolution of a scalar field \(h(\vec{x},t)\)  representing e.g. the height of an interface above point \(\vec{x}\)  on a substrate in \(\mathbb {R}^d\)  at time \(t\) , as [1]}
\(\partial _t h = \nu \nabla ^2 &h + \frac{\lambda }{2} (\nabla h)^2 + \eta (\vec{x},t) , \\\langle \eta (\vec{x},t) \eta (\vec{x}^{\prime },t^{\prime }) \rangle &= 2D \, \delta (\vec{x}-\vec{x}^{\prime }) \delta (t-t^{\prime }) ,\nonumber \) 
